What may come beyond the Wheeler-DeWitt approach?

T. P. Shestakova

REVIEW 4 major objections 4 minor 38 references

What may come beyond the Wheeler-DeWitt approach?

T0 review · 4 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The Wheeler-DeWitt equation loses its meaning once asymptotic states are abandoned, and the extended phase space approach replaces it with a gauge-dependent Schrödinger equation.

desk verdict A serious programmatic case for the extended phase space approach, but the load-bearing steps—full-gravity equivalence and loss of Wheeler-DeWitt—are asserted, not derived. read the letter →

arxiv 2607.19958 v1 pith:OATCRYVU submitted 2026-07-22 gr-qc

T. P. Shestakova This is my paper · ORCID

classification gr-qc

keywords quantumgravityWheeler-DeWittequationextendedphasespacepathintegralquantizationgaugeinvariancecosmologyreferenceframesunitarity

topics Quantum Gravity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that the Wheeler-DeWitt equation, long treated as the master equation of canonical quantum gravity, rests on assumptions that fail for a Universe with nontrivial topology. It argues that the usual derivation requires asymptotic states, whose existence contradicts the idea that quantum gravity allows all spacetime topologies; without them, gauge invariance of the path integral cannot be proven and the Wheeler-DeWitt equation loses its sense. As an alternative, the extended phase space approach keeps gauge and ghost variables as genuine canonical degrees of freedom and derives a Schrödinger equation directly from the gauge-fixed path integral. The resulting quantum state describes the geometry of the Universe as seen from a particular reference frame, so different spacetime regions can obey different quantum dynamics. This makes quantum gravity fundamentally observer-dependent and opens a natural mechanism for the breakdown of unitarity.

What carries the argument

The load-bearing mechanism is the extended phase space: gauge variables (the lapse and shift functions, or the g0µ components of the metric) and ghost fields are promoted to canonical variables, with an effective Hamiltonian built from the original Lagrangian plus a gauge-fixing term. In this setting the gauge condition and the constraint become ordinary Hamilton equations, canonical transformations that touch gauge variables are well defined, and the global symmetry of the effective action supplies a generator that reproduces the correct gauge transformations for all variables. Quantizing this system without asymptotic boundary conditions gives the mathematical Schrödinger equation and then

What would settle it

A rigorous construction of a gauge-invariant path integral for quantum gravity with nontrivial spacetime topology and no asymptotic boundary conditions—for example, by averaging over all gauge transformations—would directly contradict the claim that the Wheeler-DeWitt equation loses its sense without asymptotic states.

Watch

Extended reading notes

Core claim

The Wheeler-DeWitt equation is presented not as a law of nature but as an artifact of applying constrained Hamiltonian quantization to gravity while discarding gauge degrees of freedom and assuming asymptotic boundary conditions in the path integral. Refusing those assumptions, the paper derives a mathematical Schrödinger equation for the extended phase-space wave function, then, using the gauge condition as a quantum constraint, a physical Schrödinger equation whose Hamiltonian depends explicitly on the chosen gauge condition. The wave function gives the probability amplitude for the geometry of the Universe as it appears to an observer in the reference frame fixed by that gauge condition.

Load-bearing premise

The derivation assumes that the extended phase-space Hamiltonian dynamics, which the paper states has been proven equivalent to the gauge-fixed Lagrangian dynamics only for finite-dimensional models and for the spherically symmetric gravitational case, carries over unchanged to the full gravitational field.

Editorial extensions

If this is right

The Wheeler-DeWitt equation is replaced by a Schrödinger equation, so approaches built exclusively on the Wheeler-DeWitt equation do not exhaust the possibilities for quantum gravity.
The quantum state of the Universe is reference-frame dependent; it must be described relative to a physical observer rather than as a single gauge-invariant object.
In spacetime regions described by different gauge conditions, different physical Hamiltonians act, and transitions between the corresponding bases can violate unitarity at reference-frame boundaries.
Quantum gravitational corrections to cosmological perturbations and the cosmic microwave background obtained from the physical Schrödinger equation differ from those obtained in the semiclassical Wheeler-DeWitt framework.
A nontrivial spacetime topology can be covered piecewise by different local quantum descriptions, each with its own reference frame, instead of requiring one global state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the physical Schrödinger equation is genuinely gauge-dependent, quantum cosmology gains a relational notion of time: each reference frame carries its own clock, and comparing frames may require observer-dependent time transformations rather than a single universal Hamiltonian.
The mechanism offers a non-artificial source of the arrow of time: gravity-induced non-unitarity at frame boundaries would make irreversibility a geometrical effect, without introducing non-Hermitian operators by hand.
A concrete testable extension would be to compute specific CMB observables, such as the power spectrum or non-Gaussianity, from the gauge-dependent Schrödinger equation for several explicit gauge conditions; the paper shows that corrections differ but stops short of giving unambiguous observational signatures.
If unitarity breaks down generically at frame boundaries, the approach may bear on black-hole information questions: quantum information could be distributed across patches of spacetime rather than stored in horizon microstates.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take. The paper is worth a referee's time, but it is not a self-contained research paper; it's a well-written advocacy piece for a program the author has been developing for years. The reader's conditional verdict and the stress-test note are roughly right: the central gaps are real, and they're stated in the paper itself.

What it does well: it lays out the known problems with Dirac quantization and the Wheeler-DeWitt equation in a clear order, especially the non-canonicity of the ADM transformation and the failure of constraints to generate the full 4D diffeomorphism group. The extended phase space construction gives a consistent Hamiltonian dynamics in which gauge conditions are Hamilton equations, and the Noether-based BRST generator works for all variables. That part is a legitimate resolution of issues that have been raised. The discussion of asymptotic states and reference frames is thought-provoking: if you drop asymptotic states, gauge invariance of the path integral is no longer guaranteed, and a gauge-dependent Schrödinger equation becomes plausible. The potential connection to unitarity violation at boundaries between reference frames is a genuinely interesting idea.

The soft spots are exactly where the reader put them. First, the claim that refusing asymptotic states invalidates the Wheeler-DeWitt equation is asserted, not proved. That is the conceptual linchpin, and it needs a real argument. Second, the mathematical and physical Schrödinger equations are quoted from the author's earlier papers [25-27,34,35]; they are not rederived here. Third, and most importantly, the full-gravity extension relies on an equivalence that the paper itself states has been proved only for finite-dimensional models and the spherically-symmetric case. The step from those to full 3+1 gravity is not supplied. The decomposition (31) into gauge-fixed wave functions also gets no completeness argument. These are not internal contradictions, but they are load-bearing assumptions.

If this paper comes to a journal, send it to peer review—the question of whether Wheeler-DeWitt is replaceable deserves serious scrutiny, and the paper is a coherent statement of an alternative. But referees should insist that the no-asymptotic-states argument and the full-gravity equivalence be stated as explicit assumptions or proved, and the distinct predictions should be traced back to the prior papers in enough detail that they can be checked. As it stands, it's a useful map of a program, not a proof that the program works.

Referee Report

4 major / 4 minor

Summary. The paper argues that the Wheeler-DeWitt (WdW) approach, broadly understood, is not the only viable route to quantum gravity and should be replaced by the extended phase space approach. It claims that the Dirac quantization scheme, from which the WdW equation descends, has unresolved problems: gauge variables are not genuinely canonical, the constraints do not generate the correct four-dimensional gauge transformations, and path-integral derivations of the WdW equation require asymptotic boundary conditions that are unjustified when spacetime topology is allowed to be non-trivial. The alternative presented is based on the Faddeev–Popov effective action with gauge conditions written in differential form, so that gauge degrees of freedom become canonical and the BRST charge can be constructed by the Noether theorem. Quantizing this extended phase space path integral yields a 'mathematical' Schrödinger equation, which, after a delta-function gauge-fixing decomposition, becomes a 'physical' Schrödinger equation whose Hamiltonian depends on the chosen reference frame. The paper concludes that this genuinely goes beyond the WdW approach, with different predictions for cosmological perturbations and possible unitarity violation at boundaries between reference frames. The central technical claims, however, rest on unproved extrapolations to full gravity and on the author's prior papers.

Significance. If the extended phase space program is correct, it would offer a concrete alternative to the WdW equation with falsifiable consequences, especially for cosmological perturbations and for unitarity at transitions between reference frames. The paper identifies real and often-discussed limitations of the Dirac/WdW treatment: the non-canonical status of gauge variables, the mismatch between constraint-generated and gauge transformations, and the role of asymptotic states in deriving the WdW equation. The author is also honest about the restricted domain in which the Lagrangian–Hamiltonian equivalence has been proved. At the same time, the distinctive conclusions of the paper—the gauge-dependent physical Schrödinger equation, the difference in predictions, and the unitarity-violation claim—are not derived here for full gravity and depend substantially on unstated or prior-work results. The significance is therefore conditional: the paper lays out a coherent research program and identifies a genuine gap in the standard narrative, but it does not, in this manuscript, supply the missing full-gravity derivation.

major comments (4)

[Section 3 and Eqs. (19), (27), (32)-(33)] The central derivation chain for full gravity is incomplete. The paper states that the equivalence of the extended set of Lagrangian and Hamiltonian equations 'has been proved for models with a finite numbers degrees of freedom, as well as for the spherically-symmetric (infinite dimensional) gravitational model.' No proof is given for full (3+1)-dimensional gravity. This equivalence is needed for the Hamiltonian (19), and therefore for the mathematical Schrödinger equation (27) and the physical Schrödinger equation (32)-(33). The discussion in §3.1 shows that a class of transformations is canonical in extended phase space, but that does not establish dynamical equivalence in the full theory. This is the main load-bearing gap in the paper.
[Abstract; Section 4.1] The central premise that 'if one refuses the assumption about asymptotic states, one cannot prove the gauge invariance of the path integral, and the Wheeler-DeWitt equation loses its sense' is asserted rather than proved. The text invokes Wheeler's spacetime foam and Hawking's topological considerations, but no precise statement of what is meant by gauge invariance of the path integral, no proof of its failure without asymptotic boundary conditions, and no analysis of existing WdW derivations that do not rely on asymptotic states are supplied. Since this premise is used to justify replacing WdW with a Schrödinger equation, it needs to be stated as a theorem with hypotheses or at least given a much more explicit argument.
[Section 4.2, Eqs. (30)-(33)] The transition from the classical Hamilton equation (29) to the quantum commutator (30), and the claim that the general solution of the mathematical Schrödinger equation admits the decomposition (31), are not derived. This is not a presentation issue: the physical Schrödinger equation (32)-(33), the gauge dependence of H^(phys)[f], and all subsequent predictions follow from this decomposition. The operator-ordering ambiguity and the measure M in (28) are also free parameters whose choice can affect the resulting physical Hamiltonian. The paper needs to provide a derivation of these steps from the path integral, or at least state and prove the required completeness and operator-ordering theorem.
[Section 5 and Refs. [27], [28], [34], [35]] The distinctive predictions—different quantum gravitational corrections for cosmological perturbations and possible unitarity violation at boundaries between reference frames—are quoted from the author's previous papers and are not reproduced or even stated as precise results here. Since these predictions are the evidence that the extended phase space approach is 'really beyond' the WdW approach, the reader cannot assess their validity from this manuscript. The author should either present the derivations of these predictions in an appendix or state the precise assumptions, equations, and results on which the claims rest.

minor comments (4)

[Abstract] Typo: 'more adequate then' should be 'more adequate than'.
[Title page / author affiliation] Typo: 'Souther n Federal University' should be 'Southern Federal University'.
[References [17] and [16]] The name 'Becchi, Roust, Stora' appears to be a typo for 'Becchi, Rouet, Stora'; similarly, 'M. Hennaux' should be 'M. Henneaux'.
[Section 2] The paragraph beginning 'in 1958' should be capitalized: 'In 1958...'.

Circularity Check

2 steps flagged · score 4.0 of 10

No equation-level circularity is exhibited, but the extended-phase-space Schrödinger equation and the paper's distinctive unitarity/prediction claims are imported from the author's own prior papers [25-27, 28, 34, 35]; the full-gravity equivalence is explicitly left unproved.

self citation load bearing [Section 4.2, Eqs. (27)-(28), citing Refs. [25]-[27]]
"Further generalisation was made in our papers [25, 26] for constrained systems with finite numbers of degrees of freedom, and in the recent paper [27] for systems with infinite numbers of degrees of freedom. The main result is the Schrödinger equation ... We shall refer to Eq.(27) as a mathematical Schrödinger equation, since it is a direct mathematical consequence of the path integral without asymptotic boundary conditions."

The mathematical Schrödinger equation (27) is the central replacement for the Wheeler-DeWitt equation and the basis for the physical Schrödinger equation (32). It is not derived in this paper; it is taken verbatim from prior work by the same author (plus co-authors in [27]). The 'beyond Wheeler-DeWitt' claim therefore reduces, at its crucial step, to a self-citation chain rather than to a derivation displayed here. Nothing in the present text independently reproduces or machine-checks that result, so the citation is load-bearing rather than independent.
self citation load bearing [Sections 4.3 and 5, Refs. [28], [34], [35]]
"It was demonstrated in [28], that, in general, a transition to another basis, i.e. a transition to a region with different gauge conditions, is not a unitary operation. ... These results demonstrate that quantum gravitational corrections differ depending on the main equation (the Wheeler-DeWitt or Schrödinger equation) as well as on additional assumptions, which are arbitrary enough."

The two most distinctive advertised consequences—unitarity violation on boundaries between reference frames and different quantum-gravitational corrections for cosmological perturbations—are asserted by citing the author's own earlier papers [28] and [34,35]. They are not re-derived in the present work nor checked against an external benchmark. The conclusion that the extended phase space approach yields predictions beyond the Wheeler-DeWitt approach is thus carried by self-citations.

full rationale

The paper does build a substantial part of its derivation self-containedly: the extended phase-space Hamiltonian (19) is constructed from the Faddeev-Popov effective action, the BRST generator (25) is obtained from the Noether theorem, and the physical Schrödinger equation (32) follows by substituting the gauge-condition decomposition (31) into the mathematical equation (27). None of these steps is circular by construction—I found no equation that is fed back as a prediction. However, the crucial mathematical Schrödinger equation (27) is not derived in this text; it is imported from the author's own Refs. [25,26,27], and the unitarity/prediction claims of Sections 4.3 and 5 are imported from Refs. [28,34,35]. Under the instruction that self-citation is independent support only when machine-checked, code-reproduced, or externally falsifiable, these citations are load-bearing self-citations. I also flag the paper's own limitation in Section 3: the equivalence of Lagrangian and Hamiltonian dynamics 'has been proved for models with a finite numbers degrees of freedom, as well as for the spherically-symmetric (infinite dimensional) gravitational model'—the extension to full gravity is not proved there, and the subsequent Schrödinger equations for gravity inherit this gap. That is a correctness risk, not an equation-level circularity. Accordingly, the score is 4: some self-citation is load-bearing, but the central derivation still has independent content and no fitted input is renamed as a prediction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

No new particles, forces, or dimensions are introduced; the gauge variables, ghosts, and Lagrange multipliers are standard Faddeev-Popov/BFV objects. The main free choices are the gauge condition, the integration constant k, operator ordering, and the path integral measure, all of which affect the physical Schrödinger equation and its predictions.

free parameters (4)

Gauge condition f(q) (or f^μ(gνλ) in full gravity) = unspecified arbitrary function
The physical Hamiltonian and physical Schrödinger equation explicitly depend on the chosen gauge condition; different f yield different dynamics. The theory does not determine f.
Integration constant k in δ(N − f(q) − k) = continuous real label
Labels different 'reference frame' sectors in the decomposition (31); not fixed by the theory.
Operator ordering in the quantum Hamiltonian = unspecified ('up to operator ordering')
The quantum Hamiltonian (28) is defined only up to operator ordering, which can change predictions.
Path integral measure M = not specified
The measure in the Schrödinger Hamiltonian (28) is left general and determines the quantum correction V[f]; it is not derived from first principles here.

assumptions (6)

domain assumption The Lagrangian formulation is primary, and Faddeev-Popov path integral quantization with gauge conditions in differential form applies to gravity.
This is the starting point of the extended phase space construction in Section 3; it is a methodological commitment rather than a proven theorem.
domain assumption Asymptotic boundary conditions are illegitimate in quantum gravity because all spacetime topologies should be taken into account.
The paper's central critique of the Wheeler-DeWitt equation depends on this premise, stated in Section 4.1. It is asserted rather than proven.
ad hoc to paper The equivalence of Lagrangian and Hamiltonian dynamics in extended phase space, proved for finite-dimensional models and a spherically-symmetric model, extends to full gravity.
The paper explicitly notes the proof exists only for restricted cases; full gravity is assumed to behave the same way.
ad hoc to paper The Poisson bracket relation {H, N−f(q)}=0 can be replaced by the quantum commutator [H,N−f(q)]=0, and the general solution of the Schrödinger equation admits the δ-function/ghost decomposition (31).
This converts a classical first-order equation into an operator statement without addressing operator ordering or completeness of the proposed eigenbasis.
ad hoc to paper BRST invariance can be restored by adding the total derivative term S1 in Eq. (24) without changing the physical content.
The added term is a boundary term, but in a theory without asymptotic states the treatment of boundary terms is precisely what is at issue.
standard math Standard Faddeev-Popov ghosts and BRST/Noether machinery are valid for constructing the effective action and generators.
The paper uses standard quantization technology from [17,18] and the Noether theorem; this is background mathematics, not new physics.

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Cite this review

Pith. "Pith review of What may come beyond the Wheeler-DeWitt approach?." pith.science (2026). https://pith.science/paper/OATCRYVU

@misc{pith2026260719958,
  author       = {Pith},
  title        = {Pith review of: What may come beyond the Wheeler-DeWitt approach?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OATCRYVU}},
  note         = {Machine review of arXiv:2607.19958}
}

read the original abstract

The goal of this paper is to compare the Wheeler-DeWitt approach, understood in a broad sense, with an alternative one, the so-called extended phase space approach to quantization of gravity. By "the Wheeler-DeWitt approach", I mean not only quantum geometrodynamics formulated by DeWitt in his seminal paper of 1967, but also any approach to quantization of gravity based on the Wheeler-DeWitt equation in some form. Since the Wheeler-DeWitt equation is a direct consequence of the Dirac formalism, its analysis requires examination of the latter and its application to gravity. In particular, I argue that there is a contradiction between canonical quantization and the idea put forward by founders of quantum gravity that, in this theory, all possible spacetime topologies should be taken into account. The path integral approach seems to be more adequate then the canonical approach. However, to derive the Wheeler-DeWitt equation from the path integral, most authors make the assumption about asymptotic states, that again contradicts the supposition of arbitrary spacetime topology. The extended phase space formalism is entirely based on the path integral approach. Note that if one refuses the assumption about asymptotic states, one cannot prove the gauge invariance of the path integral, and the Wheeler-DeWitt equation loses its sense. In the alternative approach, one derives the Schrodinger equation instead. Thus, the extended phase space approach is really beyond the Wheeler-DeWitt approach. The features of this alternative approach are explored with special emphasis on conclusions that cannot be obtained by using the Wheeler-DeWitt quantum geometrodynamics.

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Reference graph

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Rovelli, Class

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P. A. M. Dirac, Can. J. Math. 2 (1950) 129

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L. D. Faddeev, Sov. Phys. Usp. 25 (1982) 130

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